The One-Way Communication Complexity of Submodular Maximization with Applications to Streaming and Robustness

We first study the one-way communication complexity of Max-Card-k in the presence of two players. An informal description of the model is as follows (see Section 2 for the formal definition). The first player Alice has only access to a subset \(V_A \subseteq W\) of the ground set and the second player Bob has access to \(V_B \subseteq W\) with \(V_A\cap V_B=\varnothing\). In the first phase, Alice can query the value of a submodular objective function f on any subset of her elements, then, at the end of the phase, she can send an arbitrary message to Bob based on the information that she has. Then, in the second phase, Bob gets the message of Alice and the elements of \(V_B\). He can then query f on any subset of the elements, and his objective is to produce a subset of \(V_A \cup V_B\) of size at most k that approximately maximizes f among all such subsets.

A trivial protocol that allows Bob to always output the optimal solution is for Alice to send all the elements in \(V_A\), and for Bob to then output \(\arg \max _{S \subseteq V_A \cup V_B: |S| \le k} f(S)\). Although this protocol has an optimal approximation guarantee of 1, it has a very large communication complexity since it requires Alice to send all the elements of \(V_A\), which may be as many as \(N = |W|\) elements.

A protocol of lower communication complexity is for Alice to calculate her “optimal” solution \(S_A = \arg \max _{S \subseteq V_A: |S| \le k} f(S)\) and send only those (at most k many) elements in \(S_A\) to Bob. Bob then outputs either his “optimal” solution \(S_B = \arg \max _{S \subseteq V_B: |S| \le k} f(S)\) or \(S_A\), whichever set attains the larger value. It is not hard to see that this protocol has an approximation guarantee of \(1/2\)—that is, that \(\max (f(S_A), f(S_B))\ge 1/2 \cdot \max _{S \subseteq V_A \cup V_B: |S| \le k} f(S)\) for any \(V_A, V_B \subseteq W\).

The preceding examples indicate a natural tradeoff between the amount of communication and the approximation guarantee, with the central question being to understand the optimal relationship between these two quantities. If one further restricts Alice to only query the submodular function on sets of cardinality at most k, then the hardness result in the work of Norouzi-Fard et al. [43] for streaming algorithms implies that, in any (potentially randomized) protocol with an approximation guarantee of \((1/2+\varepsilon)\), Alice must send a message of length \(\Omega (\varepsilon N/k)\). In other words, under this restriction on Alice, the two basic protocols described earlier achieve the optimal tradeoff up to lower-order terms (in this, as in previous work, we think of \(k \ll N\)).

Restricting Alice to evaluate f only on sets of cardinality at most k may appear like a mere technical assumption to make the arguments of Norouzi-Fard et al. [43] work, especially since, to the best of our knowledge, there are no known examples where querying the submodular function on infeasible sets leads to provably better results. Perhaps surprisingly, we prove this intuition wrong and give a protocol that crucially exploits the possibility to query the value of infeasible sets.

We present this protocol in Section 4.1, where we also show that we can further reduce the message size of Alice down to \(O(k \log (k)/\varepsilon)\) elements while still obtaining an approximation guarantee of \(2/3-\varepsilon\). By allowing Alice to query f on sets of cardinality larger than k, we can thus improve the approximation guarantee of \(1/2\) to \(2/3\) while still maintaining an (almost) linear-sized message in k. In Section 4.2, we further show that the guarantee of \(2/3\) is tight in the following strong sense. In any protocol that achieves a better guarantee, Alice must send a message of roughly the same size as the trivial protocol mentioned previously that achieves an approximation guarantee of 1.

The fact that we can beat the approximation guarantee of \(1/2\) using little communication in the presence of two players gives hope that a similar result may hold for many players, and more generally in the streaming model, which can be thought of as having one player per element. The definition of the p-player setting, with \(p \ge 2\), is the natural generalization of the two-player setting. Informally (again, see Section 2 for the formal definition), the i-th player receives a private subset \(V_i \subseteq W\) of the ground set, and upon reception of a message from the previous player, she computes and sends a message to the following player. Finally, the last player’s task is to output a subset of \(V_1 \cup V_2 \cup \cdots \cup V_p\) of cardinality at most k that approximately maximizes f among all such subsets.

One can observe that the streaming algorithm of Kazemi et al. [33] yields, for any integer \(p\ge 2\), a p-player protocol that has an approximation guarantee of \((1/2 - \varepsilon)\) and where each player sends a message consisting of at most \(O(k/\varepsilon)\) elements. Moreover, the obtained protocol only queries f on sets of size at most k and is thus tight with respect to such protocols (even in the two-player setting). Similar to the two-player case, the naturally arising question is whether protocols querying f on sets of cardinality larger than k can improve over this approximation guarantee. Our most technical result shows that this is not the case as p (and k) tends to infinity.

The proof of the preceding theorem is given in Section 5. It is based on a new construction of a family of coverage functions that hides the optimal solution while guaranteeing that no solution that does not contain elements from the optimal solution can provide an approximation significantly better than \(1/2\). This result immediately implies a tight hardness result in the streaming model that we explain in the next section.

One key reason for the success of communication complexity is that results for the models it motivates, which are on their own right interesting models capturing the essence of tradeoffs involving message sizes, are often widely applicable to other models of computation. This is also the case for submodular functions. As we show in the following, our results yield both new hardness and algorithmic results in the context of data streams and robustness.

Data Stream Algorithms. We first discuss the well-known and direct connection to data stream algorithms. In the data stream model, the elements of the (unknown) ground set arrive one element at a time rather than being available all at once, and the algorithm is restricted to only use a small amount of memory. A semi-streaming algorithm is an algorithm for this model whose memory size has only a nearly linear dependence on the parameter k (the output size) and at most a logarithmic dependence on the size of the ground set. The goal is to output, at the end of the stream, a subset of the elements in the stream of cardinality at most k that approximately maximizes f among all such subsets.

The first result in this setting was given by Chakrabarti and Kale [15], who described a semi-streaming algorithm for Max-Card-k with an approximation guarantee of \(1/4\). Badanidiyuru et al. [4] proposed later a different semi-streaming algorithm that provides a better approximation ratio of \((1/2 - \varepsilon)\) and maintains at most \(O(k \log (k)/\varepsilon)\) elements in memory. This memory footprint was recently improved by Kazemi et al. [33], who obtained the same approximation guarantee while only maintaining at most \(O(k/\varepsilon)\) elements in memory.

The two last algorithms share the approximation guarantee of \(1/2-\varepsilon\). This was improved to \(1-1/e - \varepsilon\) by Agrawal et al. [1], but only under the assumption that the elements of the stream arrive in a uniformly random order. In contrast, Huang et al. [30] showed that without this assumption, one cannot obtain an approximation ratio better than \(2 - \sqrt {2} \approx 0.586\) (improving over a previous inapproximability result of \(1-1/e\) due to McGregor and Vu [37]). Hence, prior to the current work, it remained an open question whether the approximation guarantee of \(1/2-\varepsilon\) obtained by the state-of-the-art algorithms is optimal.¹ However, a direct consequence of Theorem 1.3 settles this question. Specifically, any algorithm that achieves a better approximation guarantee than \(1/2\) must (up to a polynomial factor in k) essentially store all the elements of the stream.

The proof of the preceding theorem is almost immediate given Theorem 1.3 and the well-known connection between data stream algorithms and one-way communication. Thus, it is deferred to Appendix A.1.

Robust Submodular Function Maximization. The work on algorithms for Max-Card-k has been partially motivated by the desire to extract small summaries of huge datasets. In many settings, the extracted summary is also required to be robust. In other words, the quality of the summary should degrade by as little as possible when some elements of the ground set are removed. Such removals may arise for many reasons, such as failures of nodes in a network or user preferences that the model failed to account for; they could even be adversarial in nature. Recently, this topic has attracted special attention due to its importance in privacy and fairness constraints. The robust summaries enable us to remove sensitive data without incurring much loss in performance, giving us the ability to protect personal information (the right to be forgotten) and avoid biases (e.g., gender, measurement, and design biases).

The first attempts to design algorithms that generate robust summaries assumed that the summary is simply a set of size k, and the algorithm should guarantee that the value of this set is competitive against the best possible such set even when some elements are deleted (from both the ground set and the solution set). Naturally, this objective makes sense only when the number d of deleted elements is significantly smaller than k. Accordingly, Orlin et al. [44] provided the first constant (0.387) factor approximation result to this problem for \(d = o(\sqrt {k})\), and Bogunovic et al. [11] improved the restriction on number of deletions to \(d = o(k)\) while keeping the approximation guarantee unchanged.

More recent works studied a more general variant of the preceding problem where an algorithm consists of two procedures: a summary procedure and a query procedure. The summary procedure first generates a summary \(M\subseteq W\) of the ground set W with few but typically more than k elements, without knowing the elements \(D\subseteq W\) to be deleted; after this, the set D is revealed, and the query procedure returns a solution set \(S_D\subseteq M\setminus D\) with \(|S_D|\le k\). The goal is for the final output set \(S_D\) to be competitive against the best subset of size k in the ground set without D, for any (worst-case) choice of D. More formally, such a robust algorithm is said to have an approximation guarantee of \(\alpha\) ifThis problem is usually referred to as robust submodular maximization.

The state-of-the-art result for robust submodular maximization is a \((1/2-\varepsilon)\)-approximation algorithm due to Kazemi et al. [34], whose summaries contain \(O(k + d \log k/\varepsilon ^2)\) elements. This result improved over previous results by Mirzasoleiman et al. [39] and Mitrovic et al. [40]. It should be noted that all these results enjoy a semi-streaming summary procedure, and by Theorem 1.4, the approximation ratio of \(1/2 - \varepsilon\) guaranteed by some of them is basically the best possible as long as the summary procedure remains a semi-streaming algorithm.

We present the first algorithms for robust submodular maximization whose approximation guarantee is better than \(1/2\). We do this via the following theorem, which shows that one can convert most natural two-player protocols for Max-Card-k into algorithms for robust submodular maximization. The proof of this theorem is based on a technique of Mirzasoleiman et al. [39], and we defer both this proof and a fully formal statement of the theorem to Appendix A.2.

As all the protocols we use to prove our results in this article obey the natural properties required by Theorem 1.5, one can combine this theorem with Theorem 1.1 to get the following corollary.

Analogous to Theorem 1.1, one can reduce the summaries to \(O(dk \log (k) /\varepsilon)\) many elements while guaranteeing an approximation factor of \(2/3-\varepsilon\). Unfortunately, the protocol used to prove Theorem 1.1 uses exponential time, and thus Corollary 1.6 is mostly of theoretical value. Nevertheless, we show that even when requiring efficient procedures, the factor of \(1/2\) can be beaten while only using linear message size.

The last theorem is proved in Section 6. Combining this theorem with Theorem 1.5 yields the following result for robust submodular maximization.

In this section, we prove Theorem 1.1. For that purpose, let us present Protocol 1, which is a protocol for Max-Card-k in the two-player model that uses exponential computation. In this protocol, Alice finds for every \(i \in \lbrace 0, 1, \ldots\,, 2k\rbrace\) the maximum value subset \(S_i\) of \(V_A\) of size at most i and forwards all the sets she has found to Bob. Then, Bob finds the best solution over the elements that Alice has sent and \(V_B\).

It is easy to see that Protocol 1 always outputs a feasible set; moreover, the number of elements Alice sends to Bob is \(O(k^2)\) because she sends \(2k + 1\) sets of size at most \(2k\) each. Thus, to prove that Protocol 1 obeys all the properties guaranteed by the second part of Theorem 1.1, it remains to show that it produces a \(2/3\)-approximation, which is our main objective in the rest of this section.

Let us denote by \(\mathcal {O}\) a subset of \(V_A \cup V_B\) of size at most k maximizing f among all such subsets, and let \({\rm\small OPT} = f(\mathcal {O})\). Additionally, let \(M = V_B \cup \bigcup _{i = 1}^{2k} S_{i}\) be the set of elements that Bob either receives from Alice or receives directly. Note that M is also the set of elements in which Bob looks for \(\widehat{S}\). Using this notation, we can now describe the intuitive idea behind our first observation.

Our analysis of Protocol 1 is based on two sets \(S_{k - |\mathcal {O} \cap M|}\) and \(S_{2(k - |\mathcal {O} \cap M|)}\). Observe that one candidate for \(S_{k - |\mathcal {O} \cap M|}\) is the part of \(\mathcal {O}\) that Alice got and did not forward to Bob. Thus, we know that \(S_{k - |\mathcal {O} \cap M|}\) is as valuable as \(\mathcal {O} \setminus M\). The following observation formalizes this fact.

Despite the fact that \(S_{k - |\mathcal {O} \cap M|}\) is as valuable as \(\mathcal {O} \setminus M\), it is not clear to what extent the values of the two sets “overlap” (more formally, how far is the value of their union from the sum of their individual values). If the overlap is large, then this means that \(S_{k - |\mathcal {O} \cap M|}\) is a good replacement for \(\mathcal {O} \setminus M\), and thus Bob can construct a good solution by combining \(S_{k - |\mathcal {O} \cap M|}\) with \(\mathcal {O} \cap M\). In contrast, if the overlap between \(S_{k - |\mathcal {O} \cap M|}\) and \(\mathcal {O} \setminus M\) is small, then they can be combined into a single large value set, which guarantees a large value for \(S_{2(k - |\mathcal {O} \cap M|)}\). Thus, there is a tradeoff between the values of the sets \(\hat{S}\) and \(S_{2(k - |\mathcal {O} \cap M|)}\). Lemma 4.2 formally captures this tradeoff.

If \(f(\widehat{S})\) is large, then we are done. Otherwise, the previous lemma guarantees that \(S_{2(k - |\mathcal {O} \cap M|)}\) is a very valuable set. Although this set might be infeasible (unless \(|\mathcal {O} \cap M| \ge k / 2\)), its value can be exploited by adding half of this set to \(\mathcal {O} \cap M\). The following lemma gives the lower bound on \(f(\widehat{S})\) that can be obtained in this way.

We are now ready to prove the approximation guarantee of Protocol 1 (and thus complete the proof of the second part of Theorem 1.1).

To prove also the first part of Theorem 1.1, we need to reduce the number of elements forwarded from Alice to Bob by Protocol 1. This can be done by applying geometric grouping to the sizes of the sets in \(\lbrace S_i\rbrace _{i=1}^{2k}\). More precisely, Alice only forwards the sets \(S_i\) for either \(i = 0\), \(i= \lfloor (1+\varepsilon)^j \rfloor\), or \(i= 2\lfloor (1+\varepsilon)^j \rfloor\) for some integer \(0 \le j \le \log _{1+\varepsilon } k\), where \(\varepsilon\) is the parameter from the theorem. This reduces the number of elements forwarded to \(\widetilde{O}(k/\varepsilon)\), and it is not difficult to argue that the preceding analysis of the approximation ratio still works after this reduction, but its guarantee becomes worse by a factor of \(1-O(\varepsilon)\). A formal proof of this can be found in Appendix D.

In this section, we prove the impossibility result stated in Theorem 1.2. We do that by using a reduction from a problem known as the INDEX problem, which is presented in Section 2. The same section also states an impossibility result for a generalization of this problem (Theorem 3.3), which in the context of the INDEX problem implies that any protocol guaranteeing a success probability of at least \(2/3\) for this problem must have a communication complexity of at least \(n / 144\).

Our plan in this section is to assume the existence of a protocol named \({PRT}\) for Max-Card-k in the two-players model with an approximation guarantee of \(2/3 + \varepsilon\), and show that this leads to a protocol \({PRT_{\text{INDEX}}}\) for the INDEX problem whose communication complexity depends on the communication complexity of \({PRT}\). This allows us to translate the communication complexity lower bound for protocols for INDEX to a communication complexity lower bound for \({PRT}\).

Before getting to the protocol \({PRT_{\text{INDEX}}}\) mentioned earlier, let us first present a simpler protocol for the the INDEX problem, which is given as Protocol 2 and is used as a building block for \({PRT_{\text{INDEX}}}\). Protocol 2 refers to n possible objective functions that we denote by \(f_1, f_2, \ldots\,, f_n\). (Recall that n is the length of the string that Alice receives in the INDEX problem.) To define these functions, we first need to define a set of n other functions. Let \(W^{\prime } = \lbrace w\rbrace \cup \lbrace v_i \mid i\in [n]\rbrace\). For every \(i\in [n]\), we define \(g_i:2^{W^{\prime }} \rightarrow {\mathbb {R}_{\ge 0}}\) as follows, where S is an arbitrary subset of \(W^{\prime }\):

The multilinear extension of \(g_i\) is the function \(G_i:[0, 1]^{W^{\prime }} \rightarrow {\mathbb {R}_{\ge 0}}\) defined by \(G_i(y) = \mathbb {E} \left[ g_i({\mathcal {R}}(y)) \right]\), where \({\mathcal {R}}(y)\) is a random subset of \(W^{\prime }\) including every element \(v \in W^{\prime }\) with probability \(y_v\), independently.³ In the context of \(G_i\), given an element \(v \in W^{\prime }\), we occasionally use the notation \(\mathbb {1}_v\) to denote the characteristic vector of the singleton set \(\lbrace v\rbrace\)—that is, the vector in \([0, 1]^{W^{\prime }}\) containing 1 in the v-coordinate and 0 in all other coordinates.

Let us now define the ground set \(W = W_A \mathbin ⊍W_B\), where \(W_A = \lbrace u_i^j \mid i\in [n] \text{ and } j\in [k-1]\rbrace\) and \(W_B = \lbrace w\rbrace\). Then, for every \(i\in [n]\), the function \(f_i:2^W \rightarrow {\mathbb {R}_{\ge 0}}\) is defined aswhere the vector \(y^S\in [0,1]^{W^{\prime }}\) is defined byand

We begin the analysis of Protocol 2 with the following lemma, which shows that the objective function this protocol passes to \({PRT}\) has all the necessary properties. The proof of this lemma is simple and technical, and thus we defer it to Appendix E.1. In a nutshell, it shows by a straightforward case analysis that \(g_i\) is non-negative, monotone, and submodular, then argues that the fact that \(g_i\) has these properties implies that \(f_i\) has them too.

Our next step is analyzing the output distribution of Protocol 2.

At this point, we are ready to present the promised algorithm \({PRT_{\text{INDEX}}}\), which simply executes \(\lceil 2\varepsilon ^{-1} \rceil\) parallel copies of Protocol 2, then outputs \(x_t = 1\) if and only if at least one of the executions returned this answer.

Using the last corollary, we can now complete the proof of Theorem 1.2.

In this section, we highlight our main ideas for constructing the family \(\mathcal {F}\) satisfying the properties of Lemma 5.1. We do so by presenting three families of coverage functions \(\mathcal {H}\), \(\mathcal {G}\), and finally \(\mathcal {F}\). Family \(\mathcal {H}\) is a natural adaptation of coverage functions that have previously appeared in hardness constructions (e.g., see [37]). We then highlight our main ideas for overcoming issues with those functions by first refining \(\mathcal {H}\) to \(\mathcal {G}\) and then by refining \(\mathcal {G}\) to obtain our final construction \(\mathcal {F}\).

To convey the intuition, we work with unweighted coverage functions. However, to provide a clean and concise technical presentation later on, we use weighted coverage functions to formally realize the construction plan described here.

The First Attempt: Family \(\mathcal {H}\). The construction of the family \(\mathcal {H}= \lbrace h_{o_1, \ldots , o_p} \mid o_1 \in W_1, \ldots , o_p\in W_p\rbrace\) is inspired by the coverage functions constructed in the NP-hardness result of Feige [26]. In those coverage functions, every element corresponds to a subset of the underlying universe of size \(|U|/p\). Furthermore, the optimal solution \(\lbrace o_1, \ldots , o_p\rbrace\) forms a disjoint cover of U, whereas any other element behaves like a random subset of the universe of size \(|U|/p\).

Inspired by this, we let \(h_{o_1, \ldots , {o_{p}}} \in \mathcal {H}\) be the coverage function where

Although the preceding definition is randomized, we assume for the sake of simplicity in this overview that the value of a subset equals its expected value. This can intuitively be achieved by selecting the underlying universe U to be large enough so as to ensure concentration. For a subset \(S \subseteq W \setminus \lbrace o_1, \ldots , o_p\rbrace\), we thus have that \(h_{o_1, \ldots , o_p}(S)\) equals the expected number of elements of U covered by \(|S|\) random subsets of cardinality \(|U|/p\). Hence,which is at least \((1-1/e) |U|\) if \(|S| =p\). This already highlights the first issue of the construction: the value gap between the optimal solution \(\lbrace o_1, \ldots , o_p\rbrace\), whose value is \(|U|\), and a solution disjoint from this optimal solution is only \(1-1/e\), and we need it to approach \(1/2\) as p tends to infinity.

The second and perhaps more significant issue is the indistinguishability. First, we can observe that the value of any subset \(S\subseteq W_1\) only depends on \(|S|\), and thus the selection of \(o_1\in W_1\) is indistinguishable when querying the submodular function restricted to \(W_1\). However, the same does not hold for \(o_2\) when querying the submodular function restricted to the set \(W_1 \cup W_2\). To see this, note that \(o_1\) and \(o_2\) are the only elements of \(W_1 \cup W_2\) whose corresponding subsets of U are disjoint. In other words, \(\lbrace o_1, o_2\rbrace\) is the unique maximizer to \(\max _{S \subseteq W_1 \cup W_2: |S| = 2} h_{o_1, \ldots , o_p}(S)\), and \(o_2\) can thus be identified by querying the submodular function on \(W_1 \cup W_2\). A natural idea for addressing this issue is to make all elements in \(W_2\), and not only \(o_2\), correspond to subsets of U that are disjoint of the subset corresponding to \(o_1\). Making this modification for all \(W_2, \ldots , W_p\) results in the refined family \(\mathcal {G}\) that we now describe. We note that a similar approach was used in the work of Kapralov [32] to guarantee indistinguishability.

The First Refinement: Family \(\mathcal {G}\). Motivated by the idea to make every element in \(W_i\) correspond to a subset of U disjoint from the subsets of \(o_1, \ldots , o_{i-1}\), we define the family \(\mathcal {G}= \lbrace g_{o_1, \ldots , o_p} \mid o_1 \in W_1, \ldots , o_p\in W_p\rbrace\) of coverage functions. Specifically, we let \(g_{o_1, \ldots , o_p} \in \mathcal {G}\) be the coverage function where

The preceding description of \(\mathcal {G}\) is given in a way that highlights the changes compared to \(\mathcal {H}\). Another equivalent definition of \(g_{o_1, \ldots , o_p}\) is that it is the coverage function where

From this viewpoint, it is clear that we now have the indistinguishability property of Lemma 5.1. Indeed, for \(i\in [p]\), the only subsets of U that depend on \(o_i\) in the preceding construction are those corresponding to elements in \(W_{i+1}, \ldots , W_p\). It follows that the value of a subset \(S \subseteq W_1 \cup \cdots \cup W_i\), which is a function of the subsets of U corresponding to the elements in S, is independent of the selection of \(o_i\).

Having verified indistinguishability, let us consider the value gap. First, note that we still have that the optimal solution \(\lbrace o_1, \ldots , o_p\rbrace\) covers the whole universe and thus has value \(|U|\). Now consider a set \(S \subseteq W \setminus \lbrace o_1, \ldots , o_p\rbrace\). It will be instructive to first consider the case when \(S = \lbrace v_1, \ldots , v_p\rbrace\) with \(v_i \in W_i\) for all \(i \in [p]\)—that is, S contains exactly one element from each of the sets \(W_i\). Abbreviating \(g_{o_1, \ldots , o_p}\) by g, we have in this case thatequalswhich in turn solves toHence, for sets \(S \subseteq W \setminus \lbrace o_1, \ldots , o_p\rbrace\) that contain one element from each \(W_i\), we have a value gap that approaches the desired constant \(1/2\) as p tends to infinity. The issue is that there are other subsets of \(W \setminus \lbrace o_1, \ldots , o_p\rbrace\) of significantly higher value. To see this, note that any element \(v\in W_1 \setminus \lbrace v_1\rbrace\) has a marginal value with respect to \(\lbrace v_1, \ldots , v_{p-1}\rbrace\) that is much higher than the marginal value of \(v_p\) with respect to the same set. In particular, the value of a subset \(W_1 \setminus \lbrace o_1, \ldots , o_p\rbrace\) of cardinality p is equivalent for functions in \(\mathcal {G}\) and \(\mathcal {H}\), and is thus at least \((1-1/e)|U|\). To overcome this issue (i.e., the fact that elements of \(W_1\) are more “valuable” than other elements), we modify the preceding construction to let the elements from different \(W_i\)’s correspond to subsets of different sizes.

The Second and Last Refinement: Family \(\mathcal {F}\). The family \(\mathcal {F}= \lbrace f_{o_1, \ldots , o_p} \mid o_1 \in W_1, \ldots , o_p\in W_p\rbrace\) is obtained from \(\mathcal {G}\) by selecting subsets of U of non-uniform sizes. Specifically, we carefully select numbers \(1=a_1 \lt a_2 \lt \cdots \lt a_p\) and make the elements of \(W_i\) correspond to subsets of U of size \(a_i\), then we let the total size of U be \(a_1 + a_2 + \cdots + a_k\).⁶ We now let \(f_{o_1, \ldots , o_p} \in \mathcal {F}\) be the coverage function where

The family \(\mathcal {F}\) satisfies the indistinguishability property of Lemma 5.1 for the exact same reasons \(\mathcal {G}\) satisfies it. We now explain how the values \(a_1, \ldots , a_p\) are selected so as to obtain the value gap. Consider a set \(S \subseteq W \setminus \lbrace o_1, \ldots , o_p\rbrace\) obeying \(S = \lbrace v_1, \ldots , v_p\rbrace\) for some choice of \(v_i \in W_i\) for every \(i \in [p]\). Abbreviating \(f_{o_1, \ldots , o_p}\) by f, we thus haveThe numbers \(a_1, \ldots , a_p\) are selected so that each term of this sum equals 1, and hence \(f_{o_1, \ldots , o_p}(S) = p\). Notice that this is in stark contrast to the functions in \(\mathcal {G}\) where the contributions to (1) were highly unequal. The intuitive reason we set the numbers so that these marginal contributions are the same is that we want to prove that one cannot form a subset of \(W \setminus \lbrace o_1, \ldots , o_p\rbrace\) of cardinality at most p of significantly higher value by increasing the number of elements selected from one of the partitions \(W_i\). Formally, this is proved in Section 5.3 by considering the linear extension of a concave function at the point corresponding to such a set S that contains a single element from each \(W_i\). This allows us to upper bound the value of any subset \(W \setminus \lbrace o_1, \ldots , o_p\rbrace\) of cardinality at most p by \(p + (H_p)^2\). The value gap then follows from basic calculations (see Lemma 5.2) showing that \(f_{o_1, \ldots , o_p} (\lbrace o_1, \ldots , o_p\rbrace) = |U| = \sum _{i=1}^p a_i\) is at least \(2p - H_p\) and at most \(2p\).

We formally describe the construction of the family \(\mathcal {F}\) of weighted coverage functions on the common ground set W. Recall that the ground set is partitioned into sets \(W_1, \ldots , W_p\). Furthermore, each of these sets has cardinality n, and thus \(N = |W| = n\cdot p\).

In the intuitive description (Section 5.1), we defined the functions in \(\mathcal {F}\) to be coverage functions, where the elements correspond to random subsets of the underlying universe U. Here we will be more precise and avoid this randomness. To this end, we consider a slight generalization of weighted coverage functions that we call weighted fractional coverage functions. This is just done for convenience. In Appendix E.2.1, we show that any such function is indeed a weighted coverage function.

Recall that in a weighted coverage function, every element is a subset of an underlying universe U with non-negative weights \(a:U \rightarrow \mathbb {R}_{\ge 0}\). For fractional weighted coverage functions, apart from the non-negative weights \(a:U \rightarrow \mathbb {R}_{\ge 0}\), we also associate a function \(p_v:U \rightarrow [0,1]\) with each element \(v\in W\) with the intuition that \(p_v(u)\) specifies the “probability” that \(v\in W\) covers \(u\in U\). The value \(f(S)\) of a subset \(S \subseteq W\) of the elements is then defined byA function \(f:2^W\rightarrow \mathbb {R}_{\ge 0}\) as defined earlier is what we call a weighted fractional coverage function. Note that a weighted coverage function is simply the special case of \(\lbrace p_v: v\in W\rbrace\) taking binary values.

We are now ready to define our family \(\mathcal {F}\) of weighted fractional coverage functions, which as aforementioned is equivalent to weighted coverage functions, which in turn can be approximated by unweighted coverage functions to any desired accuracy.

The underlying universe U of the coverage functions in \(\mathcal {F}\) consists of p points \(U=\lbrace u_1,\ldots , u_p\rbrace\), where the weight \(a_{u_j} \in \mathbb {R}_{\ge 0}\), for \(j\in [p]\), will be fixed later in Section 5.2.1. For notational convenience, we use the shorthand \(a_j\) for \(a_{u_j}\) and let \(A_{\ge j} = \sum _{i=j}^p a_i\). The family \(\mathcal {F}\) now contains a weighted fractional coverage function \(f_{o_1, \ldots , o_p}\) for every \(o_1 \in W_1, \ldots , o_p \in W_p\) that is defined as follows:

Note that by interpreting the \(p_v\) functions as probabilities, the preceding definition equals the intuitive description in Section 5.1: the “hidden” optimal elements \(\lbrace o_1, \ldots , o_p\rbrace\) form a disjoint cover of the universe, and every other element in \(W_j\) corresponds to a random subset of the (now weighted) universe disjoint from the subsets corresponding to \(o_1, \ldots , o_{j-1}\). Finally, by definition, for every \(S \subseteq W,\)which can be written aswhere \(\mathbb {1}\lbrace E\rbrace\) indicates whether the event E holds.

Consider a function \(f_{o_1, \ldots , o_p} \in \mathcal {F}\). From its definition (3), it is clear that \(\lbrace o_1, \ldots , o_p\rbrace\) is an optimal solution of value \(f(\lbrace o_1, \ldots , o_p\rbrace) = f(W) = \sum _{i=1}^p a_i = A_{\ge 1} = A_{\ge 1}/a_1\), which by (6) is at least \(2p - H_p\) and at most \(2p\). The following lemma therefore implies the value gap property of Lemma 5.1.

The rest of this section is devoted to the proof of the preceding lemma. Throughout, we let \(W^{\prime } = W \setminus \lbrace o_1, \ldots , o_p\rbrace\) and denote by f the submodular function obtained by restricting \(f_{o_1, \ldots , o_p}\) to the ground set \(W^{\prime }\). By definition (see (3)), we then have for every \(S \subseteq W^{\prime }\)where \(s_i = |S \cap W_i|\). The value of a set \(S \subseteq W^{\prime }\) is thus determined by \(s_1 = |S \cap W_1|, \ldots , s_p = |S \cap W_p|\). In the following, we slightly abuse notation and sometimes write \(f(s_1, \ldots , s_p)\) for \(f(S)\) to highlight that the value only depends on the number of elements from each partition and not on the actual elements.

Now assume first that S contains exactly one element from each \(W_i\), say \(S = \lbrace v_1, \ldots , v_p\rbrace\) where \(v_i \in W_i \setminus \lbrace o_i\rbrace\) for every \(i \in [p]\). Then,Recall that we selected the weights \(a_1, \ldots , a_k\) so that each of the terms in this sum equals 1 (see Section 5.2.1). Thus, we have that the value of the set S equals p, which can also be seen from the following basic calculation:

The inequality of Lemma 5.3 thus holds in the case when S contains exactly one element from each \(W_i\). However, it turns out that such a set is only an approximate maximizer to the left-hand side of the lemma, and we need an additional argument to bound the value of any set \(S \subseteq W^{\prime }\) of cardinality at most p. We do so by defining a continuous concave version \(\widehat{F}\) of the submodular function f, which, loosely speaking, can be thought of as being a continuous extension of f. By leveraging the concavity of \(\widehat{F}\), we can obtain upper bounds through a well-chosen first-order approximation. Specifically, we consider the linear upper bound on the concave function obtained by taking its gradient at the point \(\mathbf {1} = (1, 1, \ldots , 1)\) corresponding to sets that contain exactly one element of each \(W_i \setminus \lbrace o_i\rbrace\) (see (10)).

To define \(\widehat{F}\), let us first define \(F:\mathbb {R}_{\ge 0}^p \rightarrow \mathbb {R}_{\ge 0}\) to be the following continuous proxy for f:By definition, \(F(s_1, \ldots , s_p) = f(s_1, \ldots , s_p)\) for integral vectors, and thusTo simplify calculations, we further upper bound F by the function \(\widehat{F}\) defined as follows:Note that \(\widehat{F}\) is a sum of concave functions, and it is thus a concave function on its own. If we let \(D = \lbrace (s_1, \ldots , s_p) \in \mathbb {R}_{\ge 0}^p : \sum _{i=1}^p s_i = p\rbrace\) be the “feasible” region, then because of concavity,for every vector \(\mathbf {\bar{s}} = (\bar{s}_1, \ldots \bar{s}_p)\). We select \(\mathbf {\bar{s}} = \mathbf {1} = (1 \ldots , 1)\) to be the all-ones vector, which thus gives the upper boundWe now consider each of these two terms, starting with \(\widehat{F}(\mathbf {1})\). As \(\frac{a_p}{A_{\ge p}} = 1\),We have already shown that \(f(\mathbf {1}) = f(1,1,\ldots , 1) = p,\) and as F equals f on integral values, we have \(\widehat{F}(\mathbf {1}) = p\).

It remains to bound \(\max _{\mathbf {s} \in D} \langle \nabla \widehat{F}(\mathbf {1}), \mathbf {s} - \mathbf {1} \rangle\). We havewhere the inequality follows from the bounds on the partial derivatives given by Claim 5.4. Assuming that claim, we have thus shown the statement of Lemma 5.3—that is, that any solution of cardinality at most p without any optimal elements has value at most \(p + (H_p)^2\).

The claim follows from basic calculations and the identities of Lemma 5.2. As these calculations are mechanical and not very insightful, they can be found in Appendix E.2.2.

The second key property of our family \(\mathcal {F}\) is indistinguishability: if one can only query the submodular function \(f_{o_1,\ldots , o_p}\) on \(W_1 \cup W_2 \cup \cdots \cup W_{\ell }\), then no information can be obtained about which element in \(W_\ell\) is selected to be \(o_{\ell }\). Although this is intuitively clear from the description of \(\mathcal {F}\) in Section 5.1, the following lemma gives the formal proof of the indistinguishability property of Lemma 5.1.

In this section, we present our reduction using the family \(\mathcal {F}\) of (weighted) coverage functions guaranteed by Lemma 5.1. The arguments are quite similar to those of Section 4.2, but instead of reducing from the INDEX problem, we reduce from the multiplayer version CHAIN_p(n), referred to in Section 3.

Recall that in the CHAIN_p(n) problem, there are p players \(P_1, \ldots , P_p\). For \(i=1, \ldots , p-1\), player \(P_i\) has as input a bit string \(x^{i} \in \lbrace 0,1\rbrace ^n\) of length \(n,\) and for \(i=2, \ldots , p,\) player \(P_i\) (also) has as input an index \(t^i\in \lbrace 1, \ldots , n\rbrace\), where we use the convention that the superscript of a string/index indicates the player receiving it. The players are promised that either \(x^{i}_{t^{i+1}} = 0\) for all \(i=1, \ldots , p-1\) (the 0-case) or \(x^{i}_{t^{i+1}} = 1\) for all \(i = 1\, \ldots , p-1\) (the 1-case). The objective of the players is to decide whether the input instance belongs to the 0-case or the 1-case.

For convenience, we restate the hardness result of CHAIN_p(n) here (recall that the considered protocols are one-way protocols as defined in Section 3).

Theorem 3.3. For any positive integers n and \(p\ge 2\), any (potentially randomized) protocol for CHAIN_p(n) with success probability of at least \(2/3\) must have a communication complexity of at least \(n/(36 p^2)\).

Our plan in the rest of this section is to reduce the CHAIN_p(n) problem to the p-player submodular maximization problem on (weighted) coverage functions on a ground set of cardinality \(N=p\cdot n\). As n is clear from context, we simplify notation and refer to the CHAIN_p(n) problem as the CHAIN_p problem.

Let \(PRT\) be a protocol for the p-player submodular maximization problem on weighted coverage functions subject to the cardinality constraint \(k=p\). Further assume that \(PRT\) has an approximation guarantee of \(\frac{p + (H_p)^2}{2p - H_p}(1 +\varepsilon)\) for \(\varepsilon \gt 0\). We show in the following that this leads to a protocol \(PRT_{\mbox{CHAIN}_p}\) for the \(CHAIN_p{({n})}\) problem whose message size depends on the message size of \(PRT\). Since Theorem 3.3 lower bounds the message size of any protocol for CHAIN_p(n), this leads to a lower bound also on the message size of \(PRT\).

In our reduction, we use the weighted coverage functions in family \(\mathcal {F}\) whose existence is guaranteed by Lemma 5.1. These functions are defined over a common ground set W that is partitioned into sets \(W_1, \ldots , W_p\) of cardinality n each. For future reference, we letand we have \(|W| = N = p \cdot n\).

Before getting to the protocol \(PRT_{\mbox{CHAIN}_p}\) mentioned earlier, we first present a simpler protocol for the CHAIN_p problem that is used as a building block for \(PRT_{\mbox{CHAIN}_p}\). The simpler protocol is presented as Protocol 3. In other words, if the CHAIN_p instance is \(x^{1}, x^{2}, \ldots , x^{p-1}, t^2, t^3 \ldots , t^{p}\), Protocol 3 simulates \(PRT\) on the following p-player Max-Card-k instance with \(k=p\):

The last player of Protocol 3 then decides between the 0-case and 1-case depending on the value of the solution \(S \subseteq V_1 \cup V_2 \cup \cdots \cup V_p\) outputted by the last player of \(PRT\). Note that this value is informative because the elements \(\lbrace o_1, \ldots , o_p\rbrace\) are in \(V_1 \cup V_2 \cup \cdots \cup V_p\) if and only if the given CHAIN_p instance is in the 1-case.

Some care has to be taken to make sure that the i-th player of Protocol 3 can answer the oracle queries made during the simulation of player \(P_i\) of \(PRT\). This is the reason the message from the \((i-1)\)-th player to the i-th player of Protocol 3 also contains the indices \(t^2, t^3, \ldots , t^{i-1}\). Indeed, as player i also receives index \(t^i\), she can then infer the selection of \(o_1, \ldots , o_{i-1}\), which in turn allows her to calculate the value \(f_{o_1, \ldots , o_p}(S)\) of any set \(S\subseteq W_1\cup \cdots \cup W_i\) due to the indistinguishability property of Lemma 5.1.

Our next step is analyzing the output distribution of Protocol 3.

The assumption that \(PRT\) has an approximation guarantee of \(\frac{p + (H_p)^2}{2p - H_p}(1 +\varepsilon)\) implies the following success probability in the 1-case.

At this point, we are ready to present the promised protocol \(PRT_{\mbox{CHAIN}_p}\), which simply executes \(\lceil 2\varepsilon ^{-1} \rceil\) parallel copies of Protocol 3, and then determines that the input was in the 1-case if and only if at least one of the executions returned this answer.

Using the last corollary, we can now complete the proof of Theorem 1.3.

We consider the simple deterministic protocol for the two-player setting given as Protocol 4. Without loss of generality, we assume that \(|\mathcal {O} |=k\), \(|V_A|\ge 2k\), and \(|V_B|\ge k\), as otherwise these properties can easily be obtained by adding dummy elements of value zero to \(V_A\) and \(V_B\) (and consequently, also to W), and complementing \(\mathcal {O}\) with such elements to make sure that \(|\mathcal {O} |=k\).

Our main result here is that Protocol 4 has an approximation factor that is strictly better than \(1/2\).

Our focus here is on highlighting some key arguments explaining why one can beat the factor of \(1/2\). Thus, we keep our analysis simple instead of aiming for the best approximation ratio achievable.

We define \(k_A= |\mathcal {O}\cap V_A|\), \(k_B = |\mathcal {O}\cap V_B|\), and let \(\lbrace a_1,a_2,\ldots , a_{2k}\rbrace \subseteq V_A\) be the \(2k\) elements computed by Alice—that is, they are the first \(2k\) elements chosen by Greedy when maximizing f over \(V_A\), numbered according to the order in which they were chosen. For convenience, we define the following notation for prefixes of \(\lbrace a_1, a_2,\ldots , a_{2k}\rbrace\). For any \(i\in \lbrace 0,\ldots , 2k\rbrace\), letIn particular, \(G_A^0=\varnothing\). Finally, we denote by \(G_B\subseteq V_B\) the set \(Q_{k_A}\) computed by Bob—that is, this is a Greedy solution of the problem \(\max \lbrace f(S): S\subseteq V_B, |S|\le k_B\rbrace\). To show Theorem 6.1, we prove that one of the two sets \(X_{k_A}\) or \(Y_{k_A}\) leads to the desired guarantee—that is,Without loss of generality, we assume that \(f(\mathcal {O} \cap V_A) \gt 0\) and \(k_A \gt 0\). Otherwise, \(f(\mathcal {O}) = f(\mathcal {O} \cap V_B)\) and the optimal value can be achieved with a set fully contained in Bob’s elements \(V_B\). Hence, by the approximation guarantee of Greedy, we get \(f(Y_{k_B}) \ge (1-e^{-1})\cdot f(\mathcal {O})\), and (13) clearly holds. A key quantity that we use in our analysis is the following:In other words, \(\Delta _A\) is a normalized way (normalized by \(f(\mathcal {O} \cap V_A)\)) to measure by how much the greedy solution \(G_A\) would further improve when adding all elements of \(\mathcal {O} \cap V_A\) to it. Notice that the monotonicity and submodularity of f guarantee together \(f(\mathcal {O} \cap V_A \mid G_A) \in [0, f(\mathcal {O} \cap V_A)]\), and thus, after the normalization, we get \(\Delta _A\in [0,1]\). As we show in the following, we can exploit both small and large values of \(\Delta _A\) to improve over the factor \(1/2\). To build up intuition, consider first the following ostensibly natural candidate for a hard instance. Assume thatis a submodular maximization instance with maximizer \(\mathcal {O} \cap V_A\) and the Greedy solution, which is \(G_A\), has value \(f(G_A)\) very close to \((1-e^{-1})\cdot f(\mathcal {O} \cap V_A)\), which is the worst-case guarantee for Greedy. By looking into the analysis of Greedy, this happens only when \(\Delta _A\) is very close to \(e^{-1}\). However, it turns out that in this ostensibly bad case, our algorithm is even about \((1-e^{-1})\)-approximate due to the following. The small value of \(\Delta _A \approx e^{-1}\) implies that \(G_A\) and \(\mathcal {O} _A\) behave similarly in terms of how the submodular value changes when adding elements from \(V_B\). This is important to make sure that Bob can complement \(G_A\) to a strong solution through adding elements from \(V_B\). More precisely, adding \(\mathcal {O} \cap V_B\) to \(G_A\) increases the submodular value bywhere the first inequality follows by submodularity and the second one by monotonicity of f. Finally, if we have, as discussed, that (14) is close to a worst-case instance in terms of approximability, then \(f(G_A)\approx (1-e^{-1})\cdot f(\mathcal {O} \cap V_A)\) and \(\Delta _A\approx e^{-1}\), and henceThus, when augmenting \(G_A\) with \(k_B\) elements of \(V_B\) through Greedy, the increase \(f(X_{k_A}\mid G_A)\) in submodular value is at least around \((1-e^{-1})\cdot (f(\mathcal {O})-f(\mathcal {O} \cap V_A))\). Together with the fact that \(f(G_A)\approx (1-e^{-1})\cdot f(\mathcal {O} \cap V_A)\), this implies \(f(X_{k_A}) \gtrapprox (1-e^{-1})\cdot f(\mathcal {O})\). Hence, our protocol computed a set \(X_{k_A}\) that is even close to a \((1-e^{-1})\)-approximation for this case. The preceding example highlights that small values for \(\Delta _A\)—and this includes the worst-case value of \(\Delta _A=e^{-1}\) for a classical submodular maximization problem with a cardinality constraint—allow Bob to complement \(G_A\) in a strong way. To complete the preceding intuitive reasoning to a full formal proof, we proceed as follows. We first quantify in Lemma 6.2 the performance of Greedy on Alice’s side depending on the parameter \(\Delta _A\). This will particularly imply that \(\Delta _A \approx e^{-1}\) is indeed the worst case if the task is for Alice to select \(k_A\) elements of highest submodular value. It also quantifies how Greedy, run on Alice’s side, improves for values of \(\Delta _A\) bounded away from \(e^{-1}\). We then generalize and formalize in Lemma 6.4 the preceding discussion, done for \(\Delta _A\approx e^{-1}\), to arbitrary \(\Delta _A \in [0,1]\). This shows that our protocol has a good approximation guarantee whenever \(\Delta _A\) is small, and also covers the case when \(f(\mathcal {O} \cap V_A)\) is small. Finally, Lemma 6.5 covers the case when both \(\Delta _A\) and \(f(\mathcal {O} \cap V_A)\) are large. In this case, the set \(G_A\cup (\mathcal {O} \cap V_A)\)—which may have up to \(2k_A\) elements and is thus not necessarily feasible—has large submodular value. This is useful to show that \(f(Y_{k_A})\) is large due to the following. Recall that \(Y_{k_A}\) is constructed by first applying Greedy to Bob’s elements to select \(k_B\) elements \(Q_r\) and then complement \(Q_r\) with elements from \(\lbrace a_1,\ldots , a_{2k}\rbrace\). Knowing that \(G_A \cup (\mathcal {O} \cap V_A)\) has large submodular value implies a significant increase in submodular value when adding to \(Q_r\) the highest-valued elements of \(\lbrace a_1,\ldots , a_{2k}\rbrace\). Notice that this reasoning relies on the fact that Alice considers sets of cardinality larger than k.

We now show how the intuitive discussion for the case \(\Delta _A \approx e^{-1}\) can be made formal and be generalized to arbitrary \(\Delta _A\in [0,1]\).

We now show that if both \(\Delta _A\) and \(f(\mathcal {O} \cap V_A)\) are large, then there are (possibly infeasible) sets on Alice’s side of very large submodular value, which can be exploited to complement a greedy solution on Bob’s side, leading to a set \(Y_{k_A}\) with high submodular value.

Finally, the approximation factor claimed by Theorem 6.1 is obtained by combining the lower bounds provided by Lemmas 6.4 and 6.5 to bound \(\max \lbrace f(X_{k_A}),f(Y_{k_A})\rbrace /f(\mathcal {O})\). To this end, we compute the worst-case value of the two lower bounds for all possibilities of \(\Delta _A\in [0,1]\) and \(\frac{f(\mathcal {O} \cap V_A)}{f(\mathcal {O})} \in [0,1]\). This can be captured through the following nonlinear optimization problem, where, for brevity, we use x for \(\Delta _A\) and y for \(f(\mathcal {O} \cap V_A)/f(\mathcal {O})\).Thus, the optimal value of (23) is a lower bound on the approximation ratio of Protocol 4. Together with the following statement, this completes the proof of Theorem 6.1.

One easy way to do a quick sanity check of Lemma 6.6 is by solving (23) via standard numerical optimization methods, which is not difficult due to the fact that the problem has only three variables and only two non-trivial constraints (which are smooth). Nevertheless, we formally prove Lemma 6.6 by providing an analytical description of the unique optimal solution of (23) through the use of the necessary Karush-Kuhn-Tucker optimality conditions. Because this is a standard approach for deriving optima, we defer the proof of Lemma 6.6 to Appendix E.3.

We begin this section with a formal restatement of Theorem 1.5.

Observe that the properties required from \(\mathsf {P}\) by Theorem 1.5 imply that \(\mathsf {P}\) completely ignores the sets W, \(W_A\), and \(W_B\) even though these sets are part of the global information in the formal description of the two-player model in Section 2. Thus, the algorithm \(\mathcal {A}\) we design may use \(\mathsf {P}\) without specifying these sets. We give in the following, as Algorithm 1, the algorithm \(\mathcal {A}\). The design of this algorithm is based on a technique of Mirzasoleiman et al. [39]. In a nutshell, the algorithm uses \(d + 1\) independent copies of the protocol \(\mathsf {P}\) and manages to guarantee that one of them gets exactly the elements that have not been deleted.

We begin the analysis of Algorithm 1 with the following technical observation.

The last corollary implies that Algorithm 1 can indeed find a value \(\ell\) with the promised properties, and therefore it is well defined. One can also observe that the summary produced by Algorithm 1 consists of \(d + 1\) messages of \(\mathsf {P}\), and thus the number of elements in this summary is \(d + 1 = O(d)\) times the communication complexity in elements of \(\mathsf {P}\). Finally, it is clear from the description of Algorithm 1 that this algorithm runs in polynomial time when the algorithms of Alice and Bob in \(\mathsf {P}\) run in polynomial time. Hence, to prove Theorem 1.5, it only remains to argue that Algorithm 1 always outputs a feasible solution and that it inherits the approximation guarantee of \(\mathsf {P}\). From this point on, we denote by \(\widehat{S}\) the output set of Algorithm 1 and by \(V_A\) and \(V_B\) the input sets of Alice and Bob of \(\mathsf {P}_\ell\), respectively. Using this notation, we get the following observation.

We now get to analyzing the approximation ratio of Algorithm 1.

We now would like to prove Lemma 6.6. However, since it is more convenient for our proof technique, we prove a slightly stronger version of this lemma, stated in the following, where the variables y can take values within \([0,10]\) instead of just \([0,1]\).